In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.

In the slides, I am tentatively announcing some very recent (and not yet fully written up) work of Ben Green and myself establishing the Gowers inverse conjecture in finite fields in the special case when the function f is a bounded degree polynomial (this is a case which already has some theoretical computer science applications). I hope to expand upon this in a future post. But I will describe here a neat trick I learned at the conference (from the FOCS submission of Bogdanov and Viola) which uses majority voting to enhance a large number of small independent correlations into a much stronger single correlation. This application of majority voting is widespread in computer science (and, of course, in real-world democracies), but I had not previously been aware of its utility to the type of structure/randomness problems I am interested in (in particular, it seems to significantly simplify some of the arguments in the proof of my result with Ben mentioned above); thanks to this conference, I now know to add majority voting to my “toolbox”.

Ben Green and I have just uploaded our paper “The quantitative behaviour of polynomial orbits on nilmanifolds” to the arXiv (and shortly to be submitted to a journal, once a companion paper is finished). This paper grew out of our efforts to prove the Möbius and Nilsequences conjecture MN(s) from our earlier paper, which has applications to counting various linear patterns in primes (Dickson’s conjecture). These efforts were successful – as the companion paper will reveal – but it turned out that in order to establish this number-theoretic conjecture, we had to first establish a purely dynamical quantitative result about polynomial sequences in nilmanifolds, very much in the spirit of the celebrated theorems of Marina Ratner on unipotent flows; I plan to discuss her theorems in more detail in a followup post to this one.In this post I will not discuss the number-theoretic applications or the connections with Ratner’s theorem, and instead describe our result from a slightly different viewpoint, starting from some very simple examples and gradually moving to the general situation considered in our paper.

To begin with, consider a infinite linear sequence in the unit circle , where . (One can think of this sequence as the orbit of under the action of the shift operator on the unit circle.) This sequence can do one of two things:

If is rational, then the sequence is periodic and thus only takes on finitely many values.

If is irrational, then the sequence is dense in . In fact, it is not just dense, it is equidistributed, or equivalently that

We thus see that infinite linear sequences exhibit a sharp dichotomy in behaviour between periodicity and equidistribution; intermediate scenarios, such as concentration on a fractal set (such as a Cantor set), do not occur with linear sequences. This dichotomy between structure and randomness is in stark contrast to exponential sequences such as , which can exhibit an extremely wide spectrum of behaviours. For instance, the question of whether is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether is normal base 10.

Intermediate between linear sequences and exponential sequences are polynomial sequences, where P is a polynomial with coefficients in . A famous theorem of Weyl asserts that infinite polynomial sequences enjoy the same dichotomy as their linear counterparts, namely that they are either periodic (which occurs when all non-constant coefficients are rational) or equidistributed (which occurs when at least one non-constant coefficient is irrational). Thus for instance the fractional parts of are equidistributed modulo 1. This theorem is proven by Fourier analysis combined with non-trivial bounds on Weyl sums.

For our applications, we are interested in strengthening these results in two directions. Firstly, we wish to generalise from polynomial sequences in the circle to polynomial sequences in other homogeneous spaces, in particular nilmanifolds. Secondly, we need quantitative equidistribution results for finite orbits rather than qualitative equidistribution for infinite orbits .

This week I was in London, attending the New Fellows Seminar at the Royal Society. This was a fairly low-key event preceding the formal admissions ceremony; for instance, it is not publicised on their web site. The format was very interesting: they had each of the new Fellows of the Society give a brief (15 minute) presentation of their work in quick succession, in a manner which would be accessible to a diverse audience in the physical and life sciences. The result was a wonderful two-day seminar on the state of the art in many areas of physics, chemistry, engineering, biology, medicine, and mathematics. For instance, I learnt

Why modern aircraft (such as the Dreamliner and A380) are now assembled using (incredibly tough and waterproofed) adhesives instead of bolts or welds, and how adhesion has been enhanced by nanoparticles;

How recent advances in machine learning and image segmentation (including graph cut methods!) now allow computers to identify and track many general classes of objects (e.g. people, cars, animals) simultaneously in real-world images and video, though not quite in real-time yet;

… and 42 other topics like this. (One strongly recurrent theme in the life science talks was just how much recent genomic technologies, such as the genome projects of various key species, have accelerated (by several orders of magnitude!) the ability to identify the genes, proteins, and mechanisms that underlie any given biological function or disease. To paraphrase one speaker, a modern genomics lab could now produce the equivalent of one 1970s PhD thesis in the subject every minute.)

Lebesgue density theorem. Let be Lebesgue measurable. Then for almost every , we have as , where |A| denotes the Lebesgue measure of A.

In other words, almost all the points x of A are points of density of A, which roughly speaking means that as one passes to finer and finer scales, the immediate vicinity of x becomes increasingly saturated with A. (Points of density are like robust versions of interior points, thus the Lebesgue density theorem is an assertion that measurable sets are almost like open sets. This is Littlewood’s first principle.) One can also deduce the Lebesgue differentiation theorem back from the Lebesgue density theorem by approximating f by a finite linear combination of indicator functions; we leave this as an exercise.

Hamiltonian PDE, such as the Schrödinger equation, which are heuristically related (via Liouville’s theorem) to measure-preserving actions of the real line (or time axis) , somewhat in analogy to how combinatorial number theory and graph theory were related to measure-preserving actions of and respectively, as discussed in the previous lecture.

(In physics, one would also insert some physical constants, such as Planck’s constant, but for the discussion here it is convenient to normalise away all of these constants.)

In this second lecture, I wish to talk about the dichotomy between structure and randomness as it manifests itself in four closely related areas of mathematics:

Combinatorial number theory, which seeks to find patterns in unstructured dense sets (or colourings) of integers;

Ergodic theory (or more specifically, multiple recurrence theory), which seeks to find patterns in positive-measure sets under the action of a discrete dynamical system on probability spaces (or more specifically, measure-preserving actions of the integers );

Graph theory, or more specifically the portion of this theory concerned with finding patterns in large unstructured dense graphs; and

Ergodic graph theory, which is a very new and undeveloped subject, which roughly speaking seems to be concerned with the patterns within a measure-preserving action of the infinite permutation group , which is one of several models we have available to study infinite “limits” of graphs.

The two “discrete” (or “finitary”, or “quantitative”) fields of combinatorial number theory and graph theory happen to be related to each other, basically by using the Cayley graph construction; I will give an example of this shortly. The two “continuous” (or “infinitary”, or “qualitative”) fields of ergodic theory and ergodic graph theory are at present only related on the level of analogy and informal intuition, but hopefully some more systematic connections between them will appear soon.

On the other hand, we have some very rigorous connections between combinatorial number theory and ergodic theory, and also (more recently) between graph theory and ergodic graph theory, basically by the procedure of viewing the infinitary continuous setting as a limit of the finitary discrete setting. These two connections go by the names of the Furstenberg correspondence principle and the graph correspondence principle respectively. These principles allow one to tap the power of the infinitary world (for instance, the ability to take limits and perform completions or closures of objects) in order to establish results in the finitary world, or at least to take the intuition gained in the infinitary world and transfer it to a finitary setting. Conversely, the finitary world provides an excellent model setting to refine one’s understanding of infinitary objects, for instance by establishing quantitative analogues of “soft” results obtained in an infinitary manner. I will remark here that this best-of-both-worlds approach, borrowing from both the finitary and infinitary traditions of mathematics, was absolutely necessary for Ben Green and I in order to establish our result on long arithmetic progressions in the primes. In particular, the infinitary setting is excellent for being able to rigorously define and study concepts (such as structure or randomness) which are much “fuzzier” and harder to pin down exactly in the finitary world.

This week I am in Boston, giving this year’s Simons lectures at MIT together with David Donoho. (These lectures, incidentally, are endowed by Jim Simons, who was mentioned in some earlier discussion here.) While preparing these lectures, it occurred to me that I may as well post my lecture notes on this blog, since this medium is essentially just an asynchronous version of a traditional lecture series, and the hypertext capability is in some ways more convenient and informal than, say, slides.

I am giving three lectures, each expounding on some aspects of the theme “the dichotomy between structure and randomness”, which I also spoke about (and wrote about) for the ICM last August. This theme seems to pervade many of the areas of mathematics that I work in, and my lectures aim to explore how this theme manifests itself in several of these. In this, the first lecture, I describe the dichotomy as it appears in Fourier analysis and in number theory. (In the second, I discuss the dichotomy in ergodic theory and graph theory, while in the third, I discuss PDE.)

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